Counting holes in a Fermi sea without diving in

Abstract: Topologists can differentiate between bagels and pretzels by simply counting holes in each bread. The number of holes, formally described by the Euler characteristic, is a topological invariant insensitive to smooth deformation of the shape and size of an object. In condensed matter physics, we study an analogue of pastry, the Fermi sea. Like bread filled with flour, Fermi sea is filled with electrons, and nature provides a variety of exotic topology, e.g. metal copper has a Fermi sea like a pretzel with 4 handles. In this talk, I will introduce physical quantities that measure the Euler characteristic of a D-dimensional Fermi sea. In particular, I will explain how entanglement in real space, captured by the multipartite mutual information, probes the topology of Fermi sea in momentum space. This generalizes the Calabrese-Cardy formula for bipartite entanglement entropy in 1+1D conformal field theory, and provides a new perspective for characterizing the connection between topology and entanglement in gapless phases. Concepts introduced here may also be useful for developing experimental probes of Fermi sea topology.